Bronson, J., Levine, J. Whitaker R., "Lattice Cleaving: Conforming Tetrahedral
Meshes of Multimaterial Domains with Bounded Quality". Proceedings of the 21st
International Meshing Roundtable (San Jose, CA, Oct 7 - 10, 2012), pp. 191–209.
Lattice Cleaving: Conforming Tetrahedral
Meshes of Multimaterial Domains with
Bounded Quality
Jonathan R. Bronson, Joshua A. Levine, and Ross T. Whitaker
Scientific Computing and Imaging Institute, Salt Lake City, UT, U.S.A.
{bronson,jlevine,whitaker}@sci.utah.edu
Summary. We introduce a new algorithm for generating tetrahedral meshes
that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds
on dihedral angles, and guarantees geometric fidelity. Moreover, the method
is combinatoric so its implementation enables rapid mesh construction. These
meshes are structured in a way that also allows grading, in order to reduce
element counts in regions of homogeneity.
Keywords: tetrahedral meshing, multimaterial, multi-label, biomedical, conformal meshing, watertight, mesh quality, bounded quality, adaptive meshing.
1 Introduction
The finite element method (FEM) is ubiquitous in the field of scientific computing when employing partial differential equations on complicated domains.
Its combination of flexibility and numerical consistency make it the method
of choice for simulations across a wide range of physical phenomena including electromagnetics, fluid dynamics, and solid mechanics. FEM relies on a
decomposition of a domain into a union of discrete elements, in the form
of a mesh. These elements conform to important geometries in the domain,
such as the interfaces between materials or boundary conditions. While FEM
allows for a wide range of grid types and topologies, in practice many implementations use tetrahedral domain decompositions because they offer a
good compromise between simplicity of mesh generation, generality, ability
to conform to complex geometries, and numerics.
With FEM, the solutions of PDEs are associated with a linear system
induced by the operator and the boundary conditions. The approximated
differential operator depends on the mesh elements, and the shapes of these
elements impact the structure of this matrix—most importantly its condition
number [5]. The condition number of the linear system, which is usually quite
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large, in turn controls the speed and/or accuracy of the numerical solution to
the linear system. Thus, a second important requirement of the underlying
mesh is the quality of the underlying elements. The dual requirements of
meshes that conform to geometry and meshes that have good quality elements
are often in conflict. Thus, meshing algorithms must make tradeoffs between
quality and geometric fidelity.
These mesh requirements also interact with the specific nature of the geometric constraints and the mechanisms by which they are specified. In this
work, we consider FEM simulation problems that specify materials volumetrically. That is, the physical materials are given by functions on the domain
that evaluate to the appropriate material at a given location, and material
interfaces are where these functions transition from one value to another.
This volumetric specification is natural in biomedical simulations based on
images [25], where material boundaries are derived from segmentations or
labels, as well as simulations that rely on implicit representations of physical
interfaces [10]. In this paper we specifically address tetrahedral meshing with
multimaterial interfaces, where the geometric constraints are non-manifold
structures with higher-order junctions of three and four materials.
Contributions: In this paper we describe a new meshing algorithm, lattice cleaving, for generating tetrahedral meshes for multimaterial domains
that are specified as a collection of continuous indicator functions. The output meshes conform approximately to interfaces between materials, including non-manifold regions where multiple materials meet. The output meshes
have tetrahedral elements with provably bounded dihedral angles as well as
a guaranteed fidelity of sufficiently large features. Lattice cleaving relies on a
regular background lattice, with a resolution determined by the user, which
is subdivided or cleaved to conform to material boundaries. For each cleaved
background tetrahedra, it applies and modifies a stencil, used to approximate the geometry while not destroying the good quality of elements in the
background lattice. Lattice cleaving requires a small, fixed number of passes
through the background grid and therefore leads to reliably fast run times.
Results on biomedical volumes and fluid simulations demonstrate the algorithm reliably achieves fast run times, geometric fidelity, and good quality
elements.
2 Related Work
The literature on unstructured 3D mesh generation is vast, partly as a
byproduct of the wide utility of these meshes to application areas in science and engineering. Here we divide the discussion along the two major
constraints on this meshing problem: (1) producing high quality elements
and (2) conforming to complex surfaces. In addition, we review lattice-based
meshing algorithms which use, in part, similar techniques to our own.
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2.1
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Boundary Conforming Mesh Generation
In the past decade, a significant amount of effort has gone into building high
quality surface meshes using Delaunay triangulations. One of the most popular strategies relies on Delaunay refinement [11, 27], which iteratively inserts
sample points on the domain boundary until conditions are met for sufficiently capturing both the topology and geometry of surfaces. These surface
meshes are typically inputs for conformal tetrahedral meshing algorithms,
with further refinements of the volumetric regions. Boissonnat and Oudot [4]
and Cheng et al. [9] pioneered the first variants on provable algorithms for
performing Delaunay refinement that capture the topology of smooth, surfaceboundary constraints. Extending these ideas to more complex, piecewisesmooth, and non-manifold domains followed [8]. However, these algorithm
rely on various strategies for protecting features on the material boundaries,
and the implementations of these schemes are challenging. Thus, simplifying
assumptions are required in the protection scheme to make them practical [12, 26].
The local, greedy strategy of Delaunay refinement schemes tend to find
suboptimal configurations for vertices. Variational meshing schemes attempt
to overcome this limitation by positioning vertices according to some global
energy function [6, 22, 30]. These strategies typically decouple, to various degrees, the vertex placement problem from the triangulation/tetrahedralization
problem. The optimizations are nonlinear and require multiple iterations on
gradient-descent-based strategies to find local minima. Thus, they are time
consuming, are sensitive to initializations and parameter tuning, and do not
provide typical criteria to establish guarantees on the quality of the output.
2.2
Tetrahedral Element Quality
While a large number of measurements and ratios are used to judge the quality of elements in meshes, in this work we focus on isotropic measures of
quality applicable to linear finite elements [29]. These qualities, while somewhat generic, have the advantage of being numerically useful for the large
class of elliptic operators that appear in FEM simulations of many physical
phenomena, such as incompressible flows, diffusion, and electric fields. While
there are many reasonable measures of tetrahedral mesh quality, we rely on
the worst-case dihedral angles (both minimum and maximum) over all tetrahedra. The distance of dihedral angles from 0◦ and 180◦ correlates with most
other common element quality measures.
Measures of quality are typically independent of element size. Adaptivity of
mesh size/resolution provides another key ingredient in the definition of element quality. Both Delaunay refinement [28] as well as variational schemes [1]
have been used to improve and adapt volumetric element quality, as well as
more greedy optimizations driven by local mesh improvements [14, 17]. Moreover, isotropic element quality is also indifferent to element orientations; i.e.
it penalizes anisotropy in all directions equally. The proposed work provides
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graded meshes with smaller elements near surfaces, but does not address the
problem of adaptivity and anisotropy directly.
2.3
Lattice-Based Meshing Approaches
A very common strategy for building meshes is to start with a high-quality
(e.g. regular) background mesh and modify it to adhere to geometric constraints. However, the problem of making a regular lattice conform to an
arbitrary surface presents some challenges, especially when tetrahedral quality is a concern. One strategy is to cut (or cleave) the cells of the input lattice
to match the surface, an idea popularized by the well known marching cubes
algorithm for isosurfacing [20]. The different configurations of surface/cell
intersections are typically represented by stencils with the appropriate topology. Several authors propose surface reconstruction with a piecewise linear
approximation of surfaces as they cut through the tetrahedra of a bodycentered cubic (BCC) lattice [3, 24], with extensions to non-manifold surfaces
using a collection of indicator functions (instead of the single scalar field for
isosurfacing). These algorithms examine indicator functions locally at each
vertex of the mesh element. Depending on which indicator is maximal, they
next label each vertex with a material, a generalization of inside/outside for
isosurfacing.
Working with lattices has advantages beyond just surfacing. An octree on
the regular lattice can be used to give adaptively sized elements [31]. Another
strategy for conforming is to warp a background lattice so that primitives
align with boundaries [15]. Molino et al. [23] use a BCC lattice coupled with
a red-green subdivision strategy, which they then optimize to conform to
the surface. That work empirically achieves good quality tetrahedra, albeit
with no proof of bounds. Labelle and Shewchuk [18] propose a combination
of lattice warping and stenciling, with appropriate rules that decide which
combination of strategies to use, based on the input data, in order to ensure
good quality. They describe a computer-assisted proof to compute quality
bounds for their isosurface stuffing algorithm. The proposed method shares
several aspects of the Labelle and Shewchuk approach. Like their algorithm,
we cut a BCC lattice to conform to a boundary mesh, and like their algorithm we rely on a threshold α to locally warp the lattice to remove short
edges and maintain high quality elements. However, instead of considering
a single smooth isosurface, the multimaterial boundary constraints present
nonsmooth and non-manifold structures. This adds considerable complexity
to the algorithm, which in the past has only been approached using additional
levels of subdivision, such as in Liu et al. [19] or dual-contouring [32]. Here we
show instead that a carefully designed stencil set combined with appropriate
rules for application provides quality guarantees for the resulting tetrahedra.
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3 Methodology
The proposed tetrahedral meshing algorithm operates on a collection of indicator functions. We sample these functions onto a body-centered cubic (BCC)
lattice. Similar to many surfacing and meshing algorithms [16, 20], we rely on
a set of stencils that capture local material configurations. We use the strategy
of Labelle and Shewchuck [18] to construct a set of rules for each background
BCC lattice tetrahedron that switch between two cleaving modes—either deforming the background BCC lattice or splitting the background tetrahedra
in order to conform to boundary surfaces.
Within this context, the multimaterial meshing problem presents several
important challenges. Unlike the isosurface case, one cannot easily restrict
the size of features, because feature size [1] will always go to zero where three
or more materials meet. The complexity within each lattice cell is also challenging. Considering only the material labels at vertices, the number of cases
is daunting. Furthermore, even if one represents indicator functions along lattice edges as linear, the number of possible interfaces passing through a single
edge grows with the number of materials, regardless of the conditions at the
vertices. Therefore, geometric and topological approximations are essential.
3.1
Indicator Functions
There are many papers on extensions of implicit surfaces or level sets to
multimaterial interfaces. Here we represent multimaterial interfaces using a
set of K-smooth, volumetric indicator functions, F = {fi |fi : V !→ #} [21,
25]. A material label i is assigned to a point x ∈ V if (and only if) fi (x) >
fj (x) ∀ j &= i. For any single material j, a continuous, inside-outside function
can be constructed as f˜j (x) = fj (x) − mini"=j (fi (x)), and the zero functions
of various materials will coincide at shared boundaries.
3.2
Background Lattice and Material Interfaces
Stenciling algorithms rely on a set of regular cells. The configuration of the interfaces
on these cells are used to generate an index
that corresponds to some predefined tessellation. We employ a BCC lattice (Fig. 1),
where each cell is composed of 8 normal or Fig. 1. The BCC lattice is comprimal cubic lattice vertices, plus a 9th dual posed of two grids of primal and
dual vertices. Each vertex is invertex in the center. In addition to the 12
cident to 14 edges, 36 faces, and
edges of a regular cubic cell, there are 8 di- 24 tetrahedra.
agonal edges connecting each dual vertex to
its cell’s primal vertices, and 6 connecting dual to dual. Fanning out from the
dual vertex are 24 lattice tetrahedra, each of which spans two lattice cells.
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For stencils to be applicable, decisions about the structure of each cell
must be strictly local and enumerable a priori. Our strategy for mapping data
onto the lattice entails several approximations. Each lattice vertex represents
a single material at that point, which is given by the indicator function with
maximum value. Ties are settled by a very small push away from a prioritized (or random) material. Any lattice edge that contains vertices with two
different labels contains a material transition, called a cut-point, or cut [18].
These edge-cuts sample a surface separating two materials.
A similar logic applies to junctions of more than two materials. A lattice
face with 3 unique material labels on its vertices must have an associated
transition point where all three materials meet. We refer to this point as
a triple-point (triple). The collection of triple-points in the domain define
curves where 3 materials meet. A lattice tetrahedron may have up to four
unique material labels. The 4 vertices, and the 4 function values associated
with the material labels on each vertex, define a single, isolated, material
transition point. We refer to this point as a quadruple-point (quadruple).
We restrict the number of material transitions
defined on a tetrahedron. Each lattice simplex
may contain at most a single transition point
matching its dimensionality: an edge may have
only a single cut, each face a single triple, and each
tetrahedron a single quadruple. These approximations are the multimaterial generalizations of the
approximation that underlies stencil-based isosurface algorithms, which ignore features that pass
between vertices. Fig. 2 illustrates how such a situation might manifest on an edge. These various Fig. 2. An edge with mamaterial interfaces are defined as the points where terials a and b maximum
on its endpoints, but with
the values of indicator functions of the materials
a third material c becomon the vertices are equal, and, in general, we as- ing maximum on the intersume these locations are given by an oracle.
val between
For an edge, these transitions lie on the line
segment connecting the two vertices. However, for triple or quadruple-points,
they could lie outside of the corresponding triangular face or tetrahedron,
respectively. In such cases, these points are projected back onto the tetrahedron, so that local stencils can apply (Fig. 3). These approximation lead to
a smoothing or removal of thin features that fall below the resolution of the
(a)
(b)
Fig. 3. Triples (a) and quadruples (b) are forced to lie within the primitive that
contains the associated edge-cuts
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grid—i.e. the exceptions to the above conditions are indicative of features
that fit between grid points, as proved in Section 4.2.
3.3
Quality Criteria
Within a lattice tetrahedron, we approximate the material interfaces as a
set of triangular facets that connect the various cut-points with the correct
topology. With no additional processing of the mesh data, there are only 5
unique topological cases, distinguishable by the number of edges that contain
a cut: 0, 3, 4, 5, or 6. It is impossible for a lattice tetrahedron to contain only
1 or 2 edge cuts. As illustrated in Fig. 4, these cases are composed of three
types of polyhedra: tetrahedra, triangular prisms, and hexahedra.
(a) 0-cut
(b) 3-cut
(c) 4-cut
(d) 5-cut
(e) 6-cut
Fig. 4. The 5 unique interface topologies determined by the number of cutpoints
present on a lattice tetrahedron
While these polyhedra admit multiple consistent tessellations, the output
tetrahedra could become arbitrarily bad (regardless of the tessellation chosen) depending on where interface points are located. Thus, we define a set
of violation conditions that characterize the configurations of interface points
that lead to bad tetrahedra. These conditions are used to decide when it is
appropriate to warp the background lattice (changing topology) and when it
is appropriate to leave a configuration intact. The conditions entail a threshold on the proximity between features, denoted α, and are expressed as a
fraction of the edge lengths on the background lattice.
(a) c-vertex (b) t-vertex (c) q-vertex
(d) t-edge
(e) q-edge
(f) q-face
Fig. 5. An interface point violates a feature if it falls within an intersection of half
spaces defined using α. Vertices can be violated by (a) cuts, (b) triples, and (c)
quads. Edges can be violated by (d) triples and (e) quadruples. Faces can only be
violated by (f) quadruples.
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There are three different ways in which an interface might be violating.
First, an interface point of any type may violate a lattice vertex. A cut violates
a lattice vertex if it lies within a distance α to it along the shared edge. As
shown in Fig. 5(a), even in 2D, no matter how you choose to tessellate a face,
there will always be an angle that is arbitrarily bad as the cut approaches the
lattice vertex. This principle extends to interface points of higher order (i.e.
triples and quadruples). Triple points can move within the 2D space interior
to a lattice face, and so their vertex violation region is a quadrilateral patch.
This patch is formed by the intersection of two half spaces. Each halfspace
is defined by connecting the point at distance α on one edge, to the opposite
lattice vertex (Fig. 5(b)). Similarly, quadruple-points can be anywhere inside
the lattice tetrahedron, so their vertex violation regions are formed by the
intersection of 3 half-spaces defined by planes. (Fig. 5(c)).
The second group of violations pertain to edges. Degenerate tetrahedra
can also arise if triple-points or quadruple-points lie too close to an edge. We
define the notion of edge violation in a manner consistent with vertex violations, similarly bounding the angles. Dividing lines are formed between each
vertex on the edge and the respective α position on the edge opposite that
vertex (Figs. 5(d,e)). Finally, a quadruple-point has one additional violation
condition, arising from its distance to adjacent faces. This violation region
for faces follows the same logic as the others (Fig. 5(f)).
3.4
Topological and Geometric Operations
The lattice cleaving algorithm uses two fundamental operations to ensure
mesh quality. A snap operation merges an interface point with another point
of lower order, collapsing the implicit edge between them in an output stencil.
This operation is performed on interface points that are in violation, ensuring
output stencil tetrahedra do not span bad angles. In conjuction, vertices are
warped spatially in order to conform to the interface surfaces.
The multimaterial case introduces some extra complexities in dealing with
the consequences of snaps and warps. The first challenge arises when a violated lattice vertex is incident to multiple lattice edges that have cuts. In
the two-material case, Labelle and Shewchuk [18] perform a single warp to a
single cut and remove the other cuts, essentially pulling them into the warped
vertex. In the multimaterial case, this is unsatisfactory, because the adjacent
cuts could be interfaces to several different materials. After warping, these
additional materials may still be present on the remaining edges. Thus, we
must ensure that all of the cuts and triples on incident edges and faces are
updated appropriately.
When an edge moves because one of its vertices is warped, any cut on that
edge must move along the interface the cut represents. In practice we use a
linear approximation to the interface surface to perform this update. If the
cut interface is of the same type as the violating cut that caused the warp, it
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199
naturally gets pulled into the snap like
the 2-material case. If the new location
of the cut is no longer on the edge (e.g.
moves off of one of the ends), we bring it
back onto the line segment at the appro(a) cut-project
(b) triple-project priate end point. In this way, the stenciling operation remains local. We call
Fig. 6. When a vertex warps (green this operation of recomputing the posiarrow), (a) cuts and (b) triples on in- tion of an interface on the warped latcident faces must be updated (blue ar- tice a projection, shown in Fig. 6. If the
row) to reflect the their new locations new cut position is violating, we peron the surfaces
form a snap to the lattice vertex, and
warp the vertex only if it has not already been warped previously. In this way, each lattice vertex undergoes at
most one warp.
If a lattice face moves because one of its vertices is warped, any triple on
that face may also move. We update its location using the same strategy as
with edges. If the triple leaves the face, we bring it back on, and followup
with appropriate snaps and warps as needed. Quadruples need no projection
unless a face moves in such a way that the quadruple falls outside of the new
tetrahedron—in which case it will be moved onto the nearest edge/face and
colocated with the corresponding cut/triple on that face.
This strict hierarchy of interface types
raises another complexity unique to the multimaterial case. Snaps may cause material interfaces to degenerate such that they violate
the hierarchy of interface types. For example, consider a face with a triple-point. If the Fig. 7. Degenerate triples or
cuts on two adjacent edges snap to the same quadruples are removed by sublattice vertex, the triple-point is now repre- sequent snaps
senting only a 2-material interface, with a
degenerate material region along the remaining line segment. To fix this degeneracy, the triple-point joins the two associated cuts at their warp destination, as in Fig. 7. A triple-point snap may also cause cuts to become
degenerate, and a quadruple-point snap may cause cuts and triples to become degenerate. The number of these cases is quite limited, and each one is
tested and corrected in a way that ensures a consistent hierarchy of features
and a consistent mesh.
3.5
Generalized Stencil
After snapping and warping, the polyhedra from the topological cases described in Section 3.3 may have collapsed into intermediate topologies. Each
such topology demands not only a valid tessellation stencil, but one that does
not permit degenerate tetrahedra when interface points are in nonviolating
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configurations. Moreover, each such stencil
must be consistent both within the lattice
tetrahedron, as well as across lattice faces
(Fig. 8). One of the contributions of this paper
is presenting a single generalized stencil that
can be used as a master stencil for all achievable
Fig. 8. Stencils for lattice topologies. Not only does this keep the probtetrahedra must be consistent lem of stenciling local (avoiding inconsistency
across faces
issues), but it also removes the requirement of
implementing and storing a large stencil table
that is prone to construction and transcription errors [13].
The generalized stencil is constructed from the most complicated topological case, the 6-cut case. An edge is formed between every pair of points that
could be snapped, ensuring the snapping procedure of Section 3.4 always simplifies the topology in a manner equivalent to a series of edge coallapses. On
each lattice face, edges star out from the triple-point to every edge-cut and
vertex on the boundary of the face. Similarly, on the interior of the lattice
tetrahedron, edges star out from the quadruple-point to every triple-point,
edge-cut, and vertex on the boundary of the lattice tetrahedron (Fig. 9). In
the regular case, this is equivalent to barycentric subdivision of the tetrahedron. This construction tessellates the lattice tetrahedron into 24 stencil
tetrahedra, each composed of a single vertex, cut, triple, and quadruple.
For every lattice tetrahedron that does not have this full complexity of
material interfaces (6-cuts), we choose vertices, cuts or triples, to have virtual material boundaries, as if they had already snapped. This allows us a
consistent way to tetrahedralize the polyhedra in the multimaterial stencils
without worrying about inconsistencies or tangles across faces between stencils, keeping the stenciling operation local.
To maintain consistency between adjacent
tetrahedra, virtual cuts are chosen first. The
remaining virtual points all cascade into place
from this decision. For a face that has no
triple-point, we label one of the three cuts as a
virtual triple-point. To maintain valid topologies, we always choose a cut that lies on an
edge that already contains a virtual cut. If
there is no virtual cut, a predetermined cut
is chosen. Finally, if a quadruple-point is not
present, we must choose a triple-point location to represent the virtual quadruple-point. Fig. 9. The generalized stenWe pick a triple-point using the same method cil is constructed from the 6-cut
a triple uses to pick a cutpoint. If there are case. Edges connect each intermultiple options, we choose the point that is face point to its associated lower
collocated with the most other points, virtual order features.
or real.
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Note that these rules for generalizing lattice tetrahedra operate once; they
are merely the mechanism for generating a consistent set of stencils. Also,
these rules for choosing arbitrary, but consistent, transitions from the 6-cut
case to all of the others produce different (but valid and good quality) tetrahedralizations of the similar cut patterns, depending on their orientation on
the BCC lattice. This mechanism for ensuring consistency is a multimaterial
alternative to the parity scheme used in the two-material case [18].
3.6
Algorithm
The full lattice cleaving algorithm proceeds as follows. Using an octree structure to reduce storage and allow grading, we first sample and label each
BCC lattice vertex. Cuts, triples and quadruples are computed for each lattice tetrahedron that has multiple materials. Any lattice tetrahedron that
is not the 6-cut case is generalized by labeling the locations of virtual cuts,
triples, and quadruples. This is followed by three phases of snapping and
warping.
In the first phase, all violated lattice vertices are identified and visited
exactly once. Any violating interfaces on incident edges, faces, or tetrahedra
are snapped to the vertex, and the vertex warps to the center of mass of
the interfaces, distributing any round-off equally. All adjacent nonviolating
interface points are projected to remain on their respective simplices. All
degeneracies are fixed with additional snaps.
In the second phase, all violated lattice edges are identified and visited
exactly once. If one or more triples or quadruples violates an edge, we snap
them to that edge. These snaps are implemented so that a lattice edge or
face may contain singular points of transitions or be a single material across
its entirety; we do not allow materials to lie on the half-edge, or half-face.
In the final phase, we address the problem of quadruple-points that are
too close to lattice faces. Using the face violation condition, we snap any
such quadruple-point to the triple on the face that was violated. Sometimes
a quadruple-point violates a face that no longer contains a triple. It may
have snapped to an edge-cut, or to a vertex. The quadruple-point always follows the triple-point, maintaining the hierarchy of features on each edge and
face. To finish, we output all stencil tetrahedra that contain 4 unique vertices, skipping over any that were removed during the warping and snapping
process.
4 Dihedral Angles and Geometric Fidelity
The algorithm described in Section 3.6 is designed to ensure that both dihedral angles are bounded and geometric distortion of input surfaces is controlled. In what follows, we sketch proofs of these properties.
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Bounded Dihedral Angles
The violation rules defined in Section 3.3 disallow vertex positions that could
lead to undesirable tetrahedra. The following proof relies on these carefully
designed rules for vertex placement, a particular set of properties in the
generalized stencil set, and their interaction with the background lattice.
There are several ways to classify types of bad tetrahedron [2, 7]. One useful
partitioning groups such tetrahedra into two sets: tetrahedra whose vertices
are nearly collinear, and tetrahedra whose vertices are nearly coplanar. This
classification includes not only tetrahedra with bad dihedral angles, but also
tetrahedra with bad solid angles (i.e. the spire). It is also useful to classify the
types of triangular faces that can occur these undesirable tetrahedra. These
triangles have vertices that are nearly collinear. While a tetrahedron may
still be badly shaped without their presence, (e.g. slivers), a tetrahedron that
contains poor quality triangles will certainly be of bad quality.
The rules comprising our algorithm make it impossible for output tetrahedra to become badly shaped (and consequently, they have bounded dihedral
angles). First, we show that background lattice tetrahedra stay of good quality. This properties induce constraints on the polyhedra of our output stencils.
Finally, we show that these constraints, combined with our violation conditions for warping and snapping, always lead to tetrahedra with bounded dihedral angles. Unlike the computational proof of Labelle and Shewchuk, which
relies on interval arithmetic and a numerical search, this approach does not
give a specific angle bound. This direct proof does, however, provide insights
into why this algorithm is successful at achieving bounded dihedral angles
and gives a foundation for modifications or extensions.
Definition 1. A dihedral angle θ is the angle between two planes.
A tetrahedron contains 6 internal dihedral angles. The dihedral angle between
triangular faces can be expressed as a function Φ : V 2 !→ # of the face unit
normals n̂1 and n̂2 :
Φ = arccos (n̂1 · n̂2 )
(1)
Definition 2. The aspect ratio, arf = a/l, for a triangular face, f , where a
the length of the shortest altitude and l is the length of the longest edge.
Definition 3. The aspect ratio, art = a/l, for a tetrahedron, t, where a the
length of the shortest altitude and l is the length of the longest edge.
For triangles, the aspect ratio goes to zero as the vertices approach collinearity. For tetrahedra, aspect ratio is a measure for how close the vertices of a
tetrahedron are to being either collinear or coplanar. It turns out that when
tetrahedra degerenate in these ways, it must be the case that there are either
dihedral angles of 0◦ , 180◦, or both. We next define a notion of $-good.
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Definition 4. For $ > 0, let θmin and θmax be the minimum and maximum
dihedral angles for all possible tetrahedra with art > $. A dihedral angle, θ, is
called $-good if and only if θmin ≤ θ ≤ θmax . Similarly, let φmin and φmax be
the minimum and maximum interior angles, respectively, of all triangles with
arf > $. We call a planar angle, φ, $-good if and only if φmin ≤ φ ≤ φmax .
Lemma 1. For a tetrahedron, t, with minimum and maximum dihedral angles θmin and θmax , art > 0 iff there exists κ > 0◦ such that κ < θmin and
θmax < 180◦ − κ. Similarly, for a triangle face f , with minimum and maximum interior angles φmin and φmax , arf > 0 iff there exists κ > 0◦ such that
κ < φmin and φmax < 180◦ − κ.
We omit the proof for brevity. Lemma 1 provides a crucial link between the
bound on aspect ratio and then bound on the minimum and maximum dihedral angles. The mechanisms of our algorithm are such that they specifically
ensure the property of $-good for tetrahedra, and in doing so, coupled with
Lemma 1. it must be the case that they have bounded dihedral angles.
Definition 5. Let p be a polyhedron subdivided into a set of polyhedra S. A
polyhedral face f from the set S is considered external if it is incident to ∂p.
Lemma 2. There exists a set of violation parameters αshort and αlong for
which all BCC lattice tetrahedra maintain $-good dihedral angles after warping
as described in Section 3.4.
Proof. Let t be a BCC lattice tetrahedron and rα be the radius of a ball
around each vertex. Each ball contains the possible set of points to which
its vertex may warp, given the violation parameters αshort and αlong (the
violation parameters for short and long edges, respectively). If rα = 0, no
warping takes place, and t has aspect ratio art = 0.866025. Because the
worst dihedral angle of a tetrahedron can be defined as a continuous function
of vertex positions, by the intermediate value theorem, there must exist an
rα for which art = $ > 0. Thus, by Definition 4, there must exist an αshort
and αlong for which the lattice tetrahedra maintain $-good dihedral angles
after warping.
*
)
Lemma 3. All stencil polyhedra with nonviolating vertices maintain $-good
dihedral angles around edges incident to at least one external face.
Proof. For every dihedral angle of a stencil polyhedron that spans an edge
incident to an external face, either one face or both faces are external. If
both faces are external, then the dihedral angle equals that of the enclosing
background polyhedron. By Lemma 2, we know this is an $-good dihedral
angle. If one face is internal and the other external, there is at least one
vertex, vi , on the internal face that is not incident to the external face. The
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J.R. Bronson, J.A. Levine, and R.T. Whitaker
only way for the dihedral angle to lose the $-good property is by moving
vi arbitrarily close to the external face, its edges, or its vertices. In each
case, a violation condition from Section 3.3 would be triggered making these
impossible.
*
)
Lemma 4. All output tetrahedra span at least two stencil polyhedron faces
that meet at $-good dihedral angles.
Proof. In general, any two faces may either both be external, one external and
one internal, or both internal. If both are external, by Lemma 3 we know that
the dihedral angle between these faces will remain $-good. If one is external
and the other internal, by the violation conditions of Section 3.3 we know
these faces are $-good The case of two internal faces is explicitly prevented,
for brevity we summarize why. There are 5 regular topological stencil cases
before snaps. Snapping can only simplify polyhedra, never creating additional
internal faces. Thus, if no tetrahedra span two internal faces of the polyhedra
in the regular topological cases, it is also the case that no tetrahedra span two
internal faces after edge collapses. Among the regular topological cases, only
the 5-cut and 6-cut cases have the potential for multiple internal tetrahedron
faces. The generalized stencil is designed specifically to avoid any tetrahedra
spanning the faces that are not guarded by violation conditions.
*
)
Lemma 5. All stencil triangles maintain $-good planar angles after snapping
and warping, as described in Section 3.4.
Proof. All output stencils are composed from only four types of vertices:
(lattice) vertices, cuts, triples, and quadruples, abbreviated v,c,t, and q, respectively. The vertices of a stencil triangle can exist in three ways. Either all
three vertices lie on the same lattice face, two vertices lie on the same lattice
face, or all three lie on unique lattice faces.
If all three vertices lie on the same lattice face, the violation conditions
for cuts and triples guard against such an aspect ratio. There are only three
sets of points that can become collinear, and our stencils specifically preclude
them: vcv, vtc, and ctc.
If two vertices lie on the same lattice face, the triangle’s aspect ratio, arf ,
can only fall below $ if the third vertex is in violation of the face containing
the other two. This includes the third vertex violating an edge of the lattice
face containing the two vertices.
Finally, if all three vertices lie on unique lattice faces, the triangle’s aspect
ratio, arf can only fall below $ if all three vertices are violating the vertex
incident to all three lattice faces.
*
)
Lemma 6. All output tetrahedron have $-good dihedral angles.
LC: Multimaterial Conforming Tetrahedral Meshes
205
Proof. Let t be an output tetrahedron. By Lemma 4, t has at least two faces
joined at an $-good dihedral angle along edge e. By Lemma 5 the triangles
incident to edge e are $-good. The existence of one good dihedral angle with
two good faces incident to it implies t must have $-good dihedral angles
everywhere, since it positions all four vertices.
*
)
Theorem 1. There exists a dihedral angle, θ∗ > 0◦ , such that the dihedral
angles of all tetrahedra are bounded from below by θ∗ and above by 180◦ − θ∗ .
Proof. By Lemmas 6 and 1.
*
)
This proof shows that the meshes from the lattice-cleaving algorithm never
degenerate, in fact Lemma 5 also ensures we produce no bad solid angles (e.g.
“spires”), despite them lacking bad dihedral angles. Moreover, in practice,
with proper choice of α, this bound, θ∗ , is significant, and empirical results
in Section 5 corroborate this fact.
4.2
Geometric Fidelity
We next make a statement about the quality of the surface approximation.
Let Σ be the interface surface, the complex of smooth surface patches where
two materials meet, as well as the associated curves where three materials
are coincident, and the points where four meet. Σ is a CW-complex, and
geometrically behaves as a piecewise-smooth complex [8]. It also has a well
defined medial axis MΣ that we define as the closure of the set of points in R3
that have at least two closest points in Σ. Each point in MΣ is the center of a
ball that meets Σ only tangentially. Using the medial axis, we can quantify of
the scale of features at each point p ∈ Σ. In particular, we define local feature
size, lfs : Σ → R, as the distance from each surface point to the medial axis.
Local feature size is well studied in smooth surfaces [1]. In our setting, local
feature size approaches zero near triple junctions, which meet nonsmoothly.
Consequently, we define the set of h-regular points, Σh = {p ∈ Σ | lfs(p) > h}
and restrict our claims to these.
Given a tetrahedron c in the mesh, we make a claim regarding its vertices.
For c let Σ|c be the restriction of Σ to c, defined as Σ ∩ c. We define an
h-regular tetrahedron c as one where the set of p ∈ Σ|c are regular.
Lemma 7. Given an h-regular tetrahedron c constructed from BCC lattice
edges which are no longer than h, any vertex v of c that is labeled as having
two materials (surface vertex) lies on Σ
Proof. Because v has two materials, it is the byproduct of a warp and snap
to bring it to the Σ. Prior to this, v underwent a sequence of operations that
depended on the cuts of edges incident to v. As long as there was only one
such cut, v only warped once, and directly to the surface as computed by our
indicator function oracle. We prove that because c is h-regular, this is always
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J.R. Bronson, J.A. Levine, and R.T. Whitaker
the case. Assume, for sake of contradiction, that there were multiple cuts,
of different materials, on edges incident to v (if the materials were identical
we preferentially pick the closest cut to move to). Without loss of generality,
assume there are two cuts of material type AB and BC, we call x and y.
Both x and y lie in the violating zone, on an edge incident to v. They are no
further apart than 2hαshort < h. This indicates that the two surface patches
defining these cuts are no further apart than h as well.
We show there must be a medial axis point within distance h of Σ|c. Consider the medial axis for any single connected material region. By definition
it is a deformation retract of this region, and in addition, it touches any point
where three materials meet. Thus within a single region the medial axis is
a single connected component which connects all triple-points. If we walk
along the line segment joining x and y we must therefore cross the medial
axis, because otherwise it would violate the above property. As a result, there
is a medial axis point at this crossing. This medial axis point must be within
distance h of v. However, because v lies on Σ|c , this violates the fact that c
is h-regular, leading to a contradiction.
In generalizing this case to when more than two cuts are adjacent v, the
same logic holds for any pair of them, which is sufficient to create the same
contradiction above.
*
)
All h-regular tetrahedron are well-behaved; in general, they act just like the
tetrahedra in the isosurface stuffing algorithm [18]. Most importantly, when
they do mesh a piece of the surface, they geometrically approximate the
surface. As with any pointwise probe, it is impossible to guarantee that there
are no tiny features missed on account of the mesh resolution being too coarse.
However, we can guarantee that for h-regular tetrahedra containing surface
patches of Σ, every point on an interface triangle representing this patch lies
close to Σ. This “one-sided” notion of distances mirrors Theorem 2 of [18]
(only the distance of every mesh vertex to Σ is bounded). When the mesh
is of fine enough resolution and each surface patch of Σ sufficiently smooth,
the distance bound for h-regular tetrahedra becomes two-sided—the claim
follows for distances from Σ to the mesh.
5 Results
Our implementation of lattice cleaving is extremely fast, requires virtually no
user interaction, and achieves bounded dihedral angles. These bounds depend
on choices of the violation parameters αlong and αshort . Because our meshes
represent volumes on both sides of each interface surface, the most appropriate parameters would seem to be those used by Labelle and Shewchuk [18]
in the double-sided surface case. However, our multimaterial violation conditions use slightly more conservative bounds to take into account the dihedral
angles near triples and quadruples. We picked the parameters αshort = 0.357
and αlong = 0.203 and found they achieved a worse case minimum angle of
LC: Multimaterial Conforming Tetrahedral Meshes
207
2.76◦ and worst case maximum angle of 175.426◦. In practice, after simulating
many hundreds of times steps of several dozen fluid simulations, corresponding to hundreds of millions of tetrahedra, we see much better angles for the
vast majority of meshes.
To illustrate some of the datasets for which lattice cleaving can be used,
we provide several examples. Fig. 10(a) shows a mesh generated from a segmented MRI scan of a human head. The algorithm completed in about 100
seconds and produced a mesh with roughly 5 million elements, all with dihedral angles between 4.33◦ and 157.98◦. Fig. 10(b) shows a mesh generated
from a similar scan of a human torso. The algorithm completed in under
a minute and produced a mesh with roughly 12 million elements, all with
dihedral angles between 5.11◦ and 159.91◦.
(a) head
(b) torso
Fig. 10. Meshes generated from medical data. (a) MRI scan of a human head.
Resolution: 264x264x264. Dihedral angles: [4.33◦ − 157.98◦ ]. ≈ 5.3 million elements. (b) MRI scan of a human torso. Resolution: 208x96x208. Dihedral angles:
[5.11◦ − 159.91◦ ]. ≈ 12.6 million elements.
1.8e+07
count
count/20
1.6e+07
Frequency
1.4e+07
1.2e+07
1e+07
8e+06
6e+06
4e+06
2e+06
0
0
(a)
20
40
60 80 100 120 140 160 180
Dihedral Angle
(b)
Fig. 11. A multiphase viscous fluid simulation. (a) Each frame uses a conforming
mesh to perform fluid physics. (b) A histogram of all the angles produced throughout the simulation.
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J.R. Bronson, J.A. Levine, and R.T. Whitaker
This work also applies to multiphase fluid simulation and animation. To
demonstrate this, we utilize the lattice cleaving algorithm in the core of a
multiphase viscous fluid simulation. Fig. 11(a) shows a rendering and cutaway view of the underlying mesh used for physics. This simulation used a
643 background lattice (primal vertices), required 8 seconds to mesh, and
produces, on average 1.2 million tetrahedra. Fig. 11(b) shows a histogram of
the dihedral angles generated from 350 simulation frames. The majority of
elements are of excellent quality, with small tails near the expected bounds.
Acknowledgement. This work was supported in part by grants from the US
National Institutes of Health/National Center for Research Resources Center for
Integrative Biomedical Computing, 2P41 RR01125523-12.
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